Learning the underlying Bayesian Networks (BNs), represented by directed acyclic graphs (DAGs), of the concerned events from purely-observational data is a crucial part of evidential reasoning. This task remains challenging due to the large and discrete search space. A recent flurry of developments followed NOTEARS[1] recast this combinatorial problem into a continuous optimization problem by leveraging an algebraic equality characterization of acyclicity. However, the continuous optimization methods suffer from obtaining non-spare graphs after the numerical optimization, which leads to the inflexibility to rule out the potentially cycle-inducing edges or false discovery edges with small values. To address this issue, in this paper, we develop a completely data-driven DAG structure learning method without a predefined value to post-threshold small values. We name our method NOTEARS with adaptive Lasso (NOTEARS-AL), which is achieved by applying the adaptive penalty method to ensure the sparsity of the estimated DAG. Moreover, we show that NOTEARS-AL also inherits the oracle properties under some specific conditions. Extensive experiments on both synthetic and a real-world dataset demonstrate that our method consistently outperforms NOTEARS.

翻译：学习与事件相关的潜在贝叶斯网络（以有向无环图表示）是因果推理中的关键环节，但该任务因庞大且离散的搜索空间而极具挑战性。近年来，NOTEARS[1]通过利用无环性的代数等式表征，将这一组合优化问题转化为连续优化问题，推动了该领域的快速发展。然而，连续优化方法在数值优化后容易得到非稀疏图结构，导致无法灵活剔除潜在诱导环的边或具有小值的虚假发现边。针对此问题，本文提出一种完全数据驱动的DAG结构学习方法，无需预设阈值处理小值。我们将该方法称为基于自适应Lasso的NOTEARS（NOTEARS-AL），通过应用自适应惩罚方法确保估计DAG的稀疏性。此外，我们证明在特定条件下NOTEARS-AL继承了Oracle性质。在合成数据集和真实数据集上的广泛实验表明，本方法始终优于NOTEARS。